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<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>set </b><font color="Maroon">r</font> = <font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font>;<br/>
<b>set </b><font color="Maroon">x</font> = <font color="Maroon">n</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">m</font>;<br/>
<b>set </b><font color="Maroon">y</font> = <font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font></span>)</span>;<br/>
<b>assume </b><a NAME="E1:1"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>then consider </b><font color="Maroon">t</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font></span>)</span> &amp; <font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">t</font> = <font color="Maroon">t</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E5:1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">n</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">m</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">m</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/>
<a NAME="E6:1"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">m</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T58">NAT_1:58</a></i>;<br/>
<a NAME="E7:1"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">m</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/>
<a NAME="E8:1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/>
<a NAME="E9:1"/><b>then </b>
<font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">t</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T56">NAT_1:56</a></i>;<br/>
<a NAME="E10:1"/><b>then </b>
<font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font>
 
<b>by </b><i>, , <a class="ref" href="nat_1.html#T55">NAT_1:55</a></i>;<br/>
<a NAME="E11:1"/><b>then </b>
<font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E12:1"/>
<font color="Maroon">n</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T52">NAT_1:52</a></i>;<br/>


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